Vieta Jumping as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research

FINDING: Vieta jumping is the elementary face of an affine Weyl group orbit structure on integer solutions of quadratic forms; the arXiv paper formalizes the infinite reduced word geometry underlying this. MATH: Vieta jumping on \(x^2 + y^2 = kxy + c\) (or \(ax^2 + bxy + cy^2 = n\)) corresponds to the action of the simple reflection \(s_i\) on the root lattice: \((x,y) \mapsto (kx - y, x)\) for \(k = \frac{x^2+y^2-c}{xy}\). This is precisely the Weyl group \(W(\widetilde{A}_1)\) (or higher \(\widetilde{A}_n\)) acting on the integer points of the quadratic form's level set. The affine Weyl group \(\widetilde{A}_n\) has Coxeter generators \(s_0, s_1, \dots, s_n\) with \(s_i^2=1\), \((s_i s_{i+1})^3=1\), and the orbit of a solution under these reflections is the full integer solution set. The paper's "minimal infinite reduced words" encode the infinite descent/ascent chains of Vieta jumps — each jump is a simple reflection, and the sequence of jumps is a reduced word in the affine Weyl

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com